I think that YANBU. Your lad needs to recuperate. Rest and taking things easy for a while is one way he can do that.
Pneumonia is a serious illness, and in a child as young as this, it's potentially life threatening.
When I had Chicken Pox aged 30, I was told that there is NO way that you catch it from a third party, you have to have contact with someone with it. I didn't know or meet anyone with CP directly, but my colleagues DC had it. I know someone else who caught it 3rd party too.
When I was in inventories, the exposure to other peoples germs/daily life DID impact on my immune system. i am not in Infant School, my immunosystem is stronger than a child in Yr1, but I still got loads of coughs, colds and sore throats.
Now that I don't do this job anymore, I am hardly ever ill.
So for all the muppets posters that rant and say you can't catch germs from others, that it won't make a difference, the OP is YABU, precious etc, get off those high horses and give them a break.
OP, write the letter, and tell them that you won't give permission to his doing extra duties for the remainder of this school year.